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The entire exterior of a large wooden cube is painted red
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Updated on: 14 Jul 2013, 10:38

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E

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55% (hard)

Question Stats:

67%(01:42) correct 33%(01:29) wrong based on 672 sessions

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The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

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Re: The entire exterior of a large wooden cube is painted red
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14 Jul 2013, 10:48

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TheGerman wrote:

The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

A. 6n^2B. 6n^2 – 12n + 8C. 6n^2 – 16n + 24D. 4n^2E. 24n – 24

Say n=3.

So, we would have that the large cube is cut into 3^3=27 smaller cubes:

Attachment:

Red Cube.png [ 79.39 KiB | Viewed 20636 times ]

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

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Re: The entire exterior of a large wooden cube is painted red
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14 Jul 2013, 10:52

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TheGerman wrote:

The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

Re: The entire exterior of a large wooden cube is painted red
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14 Aug 2013, 04:41

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Obviously bunuel's solution is mindblowing and the best approach we need in such PS questions ...But just while brainstorming trying to find the solution to this problem i reached here ...Try to visualise that all the smaller cubes which lie on the exterior face of the larger wooden cube have one or more faces painted red...rest all other cubes which lie beneath the first layer of cube wont have any faces painted red...

Now if we can visualise the situation .....we can see that if we remove the external layers of cube ..we will be left with cubes having none of their faces red coloured...and if we remove this external layers of cube we are basically removing one cube from each side symmetrically ...so we will be left with a cube having dimensions of (n-2).... so basically we will be left with (n-2)^3 cubes ... now if we want to find the number of cubes in the larger cube having one or more faces as red we can deduct (n-2)^3 from n^3...

Re: The entire exterior of a large wooden cube is painted red
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11 Dec 2013, 06:29

Bunuel wrote:

TheGerman wrote:

The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

A. 6n^2B. 6n^2 – 12n + 8C. 6n^2 – 16n + 24D. 4n^2E. 24n – 24

Say n=3.

So, we would have that the large cube is cut into 3^3=27 smaller cubes:

Attachment:

Red Cube.png

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

Answer: B.

Hi Bunuel,

As usual great explanation. I just have one question , what if we choose n to be 4 or 5. The visualization then becomes little difficult. What would then a better approach solve such question.

Re: The entire exterior of a large wooden cube is painted red
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12 Dec 2013, 04:04

davidfrank wrote:

Bunuel wrote:

TheGerman wrote:

The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

A. 6n^2B. 6n^2 – 12n + 8C. 6n^2 – 16n + 24D. 4n^2E. 24n – 24

Say n=3.

So, we would have that the large cube is cut into 3^3=27 smaller cubes:

Attachment:

Red Cube.png

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

Answer: B.

Hi Bunuel,

As usual great explanation. I just have one question , what if we choose n to be 4 or 5. The visualization then becomes little difficult. What would then a better approach solve such question.

You could apply the same logic:

Say n=5. So, we would have that the large cube is cut into 5^3=125 smaller cubes. Out of them (5-2)^3=27 little cubes won't be painted red at all and the remaining 125-27=98 will have at least one red face. Now, plug n=5 and see which one of the options will yield 98.

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Re: The entire exterior of a large wooden cube is painted red
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14 Sep 2014, 17:00

maggie27 wrote:

Hi Guys,When I try to substitute n = 4, it seems that both B and C works. Please help!

For plug-in method it might happen that for some particular number(s) more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only.
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Re: The entire exterior of a large wooden cube is painted red
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05 Mar 2015, 09:54

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

Re: The entire exterior of a large wooden cube is painted red
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05 Mar 2015, 10:05

sushi574 wrote:

Out of them only the central little cube won't be painted red at all and the remaining 26 will have at least one red face. Now, plug n=3 and see which one of the options will yield 26. Only B works: 6n^2 – 12n + 8 = 54 - 36 + 8 = 26.

how did you get the formula for: Number of cube inside = \((n-2)^{3}\)?- something you need to memorize?- i can't even visualize it...a cube has 6 faces (top/bottom, +4 around). if we assume a particular cube has 3 tiny cubes per row & column, we can conclude there will be an "innermost" tiny cube that need not be painted red (it would be buried inside the cube). how do we know that in order to find this innermost cube, the formula is: \((n-2)^{3}\)

Re: The entire exterior of a large wooden cube is painted red
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22 May 2017, 20:52

TheGerman wrote:

The entire exterior of a large wooden cube is painted red, and then the cube is sliced into n^3 smaller cubes (where n > 2). Each of the smaller cubes is identical. In terms of n, how many of these smaller cubes have been painted red on at least one of their faces?

A. 6n^2B. 6n^2 – 12n + 8C. 6n^2 – 16n + 24D. 4n^2E. 24n – 24

For this problem, pick the smallest n that will satisfy the problem. In this case, n =3 satisfies the criterion. You can then draw a cube with 27 individual pieces. Only the middle cube will be unpainted. So you want to then backsolve from the answer choices until you find 26.

Re: The entire exterior of a large wooden cube is painted red
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07 Dec 2017, 15:53

Hi All,

This question can be solved by TESTing VALUES. Let's TEST N = 3 (if you think about a standard Rubik's cube, then that might help you to visualize what the cube would look like).

So now every "outside" face of the Rubik's cube has been painted. There's only 1 smaller cube of the 27 smaller cubes that does not have paint on it (the one that's in the exact middle). Thus, 26 is the answer to the question when we TEST N=3. There's only one answer that matches....